Ergodicity and type of nonsingular Bernoulli actions

• 成果类型: Article
• 署名作者: Bjorklund, Michael; Kosloff, Zemer; Vaes, Stefaan
• 署名单位: Chalmers University of Technology; Hebrew University of Jerusalem; KU Leuven
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-01014-0
• 发表日期: 2021
• 页码: 573-625
• 关键词: maharam extensions; k-property

摘要

We determine the Krieger type of nonsingular Bernoulli actions G curved right arrow Pi(g is an element of G)({0,1},mu(g)). When G is abelian, we do this for arbitrary marginal measures mu(g). We prove in particular that the action is never of type II infinity if G is abelian and not locally finite, answering Krengel's question for G=Z. When G is locally finite, we prove that type II infinity does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II1 or III1. When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of Vaes and Wahl (Geom Funct Anal 28:518-562, 2018) by proving that a group G admits a Bernoulli action of type III1 if and only if G has nontrivial first L-2-cohomology.